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Data Mining: Concepts and Techniques Jianlin Cheng Department of Computer Science University of Missouri, Columbia Customized and Revised from Slides of the Text Book ©2006 Jiawei Han and Micheline Kamber, All rights reserved May 22, 2017 Data Mining: Concepts and Techniques 1 Chapter 7. Cluster Analysis 1. What is Cluster Analysis? 2. Types of Data in Cluster Analysis 3. A Categorization of Major Clustering Methods 4. Partitioning Methods 5. Hierarchical Methods 6. Density-Based Methods 7. Grid-Based Methods 8. Model-Based Methods 9. Clustering High-Dimensional Data 10. Constraint-Based Clustering 11. Outlier Analysis 12. Summary May 22, 2017 Data Mining: Concepts and Techniques 2 What is Cluster Analysis? Cluster: a collection of data objects Similar to one another within the same cluster Dissimilar to the objects in other clusters Cluster analysis Finding similarities between data according to the characteristics found in the data and grouping similar data objects into clusters Unsupervised learning: no predefined classes Typical applications As a stand-alone tool to get insight into data distribution As a preprocessing step for other algorithms May 22, 2017 Data Mining: Concepts and Techniques 3 Clustering: Rich Applications and Multidisciplinary Efforts Pattern Recognition Spatial Data Analysis Detect spatial clusters or for spatial mining tasks Image Processing Economic Science (especially market research) Bioinformatics (e.g. clustering gene expression data) WWW Document classification Cluster Weblog data to discover groups of similar access patterns May 22, 2017 Data Mining: Concepts and Techniques 4 Examples of Clustering Applications Marketing: Help marketers discover distinct groups in their customer bases, and then use this knowledge to develop targeted marketing programs Land use: Identification of areas of similar land use in an earth observation database Insurance: Identifying groups of motor insurance policy holders with a high average claim cost City-planning: Identifying groups of houses according to their house type, value, and geographical location Earth-quake studies: Observed earth quake epicenters should be clustered along continent faults May 22, 2017 Data Mining: Concepts and Techniques 5 Quality: What Is Good Clustering? A good clustering method will produce high quality clusters with high intra-class similarity low inter-class similarity The quality of a clustering result depends on both the similarity measure used by the method and its implementation The quality of a clustering method is also measured by its ability to discover some or all of the hidden patterns May 22, 2017 Data Mining: Concepts and Techniques 6 Measure the Quality of Clustering Dissimilarity/Similarity metric: Similarity is expressed in terms of a distance function, typically metric: d(i, j) There is a separate “quality” function that measures the “goodness” of a cluster. The definitions of distance functions are usually very different for interval-scaled, boolean, categorical, ordinal ratio, and vector variables. Weights should be associated with different variables based on applications and data semantics. It is hard to define “similar enough” or “good enough” May 22, 2017 the answer is typically highly subjective. Data Mining: Concepts and Techniques 7 Requirements of Clustering in Data Mining Scalability Ability to deal with different types of attributes Ability to handle dynamic data Discovery of clusters with arbitrary shape Minimal requirements for domain knowledge to determine input parameters Able to deal with noise and outliers Insensitive to order of input records High dimensionality Incorporation of user-specified constraints Interpretability and usability May 22, 2017 Data Mining: Concepts and Techniques 8 Chapter 7. Cluster Analysis 1. What is Cluster Analysis? 2. Types of Data in Cluster Analysis 3. A Categorization of Major Clustering Methods 4. Partitioning Methods 5. Hierarchical Methods 6. Density-Based Methods 7. Grid-Based Methods 8. Model-Based Methods 9. Clustering High-Dimensional Data 10. Constraint-Based Clustering 11. Outlier Analysis 12. Summary May 22, 2017 Data Mining: Concepts and Techniques 9 Data Structures Data matrix x11 ... x i1 ... x n1 Dissimilarity matrix May 22, 2017 ... x1f ... ... ... ... xif ... ... ... ... ... xnf ... ... 0 d(2,1) 0 d(3,1) d ( 3,2) 0 : : : d ( n,1) d ( n,2) ... Data Mining: Concepts and Techniques x1p ... xip ... xnp ... 0 10 Type of data in clustering analysis Interval-scaled variables Binary variables Nominal, ordinal, and ratio variables Variables of mixed types May 22, 2017 Data Mining: Concepts and Techniques 11 Interval-valued variables Standardize data Calculate the mean absolute deviation: sf 1 n (| x1 f m f | | x2 f m f | ... | xnf m f |) where m f 1n (x1 f x2 f ... xnf ) . Calculate the standardized measurement (z-score) xif m f zif sf Using mean absolute deviation is more robust than using standard deviation May 22, 2017 Data Mining: Concepts and Techniques 12 Similarity and Dissimilarity Between Objects Distances are normally used to measure the similarity or dissimilarity between two data objects Some popular ones include: Minkowski distance: d (i, j) q (| x x |q | x x |q ... | x x |q ) i1 j1 i2 j2 ip jp where i = (xi1, xi2, …, xip) and j = (xj1, xj2, …, xjp) are two p-dimensional data objects, and q is a positive integer If q = 1, d is Manhattan distance d (i, j) | x x | | x x | ... | x x | i1 j1 i2 j 2 i p jp May 22, 2017 Data Mining: Concepts and Techniques 13 Similarity and Dissimilarity Between Objects (Cont.) If q = 2, d is Euclidean distance: d (i, j) (| x x |2 | x x |2 ... | x x |2 ) i1 j1 i2 j2 ip jp Properties d(i,j) 0 d(i,i) = 0 d(i,j) = d(j,i) d(i,j) d(i,k) + d(k,j) Also, one can use weighted distance, 1 - Pearson correlation, or other disimilarity measures May 22, 2017 Data Mining: Concepts and Techniques 14 Binary Variables Object j 1 0 A contingency table for binary 1 a b Object i data 0 c d sum a c b d Distance measure for symmetric binary variables: Distance measure for asymmetric binary variables: Jaccard coefficient (similarity measure for asymmetric d (i, j) d (i, j) May 22, 2017 bc a bc d bc a bc simJaccard (i, j) binary variables): Data Mining: Concepts and Techniques sum a b cd p a a b c 15 Dissimilarity between Binary Variables Example Name Jack Mary Jim Gender M F M Fever Y Y Y Cough N N P Test-1 P P N Test-2 N N N Test-3 N P N Test-4 N N N gender is a symmetric attribute (not used) the remaining attributes are asymmetric binary let the values Y and P be set to 1, and the value N be set to 0 0 1 0.33 2 0 1 11 d ( jack , jim) 0.67 111 1 2 d ( jim, mary ) 0.75 11 2 d ( jack , mary ) May 22, 2017 Data Mining: Concepts and Techniques 16 Nominal Variables A generalization of the binary variable in that it can take more than 2 states, e.g., red, yellow, blue, green Method: Simple matching m: # of matches, p: total # of variables m d (i, j) p p May 22, 2017 Data Mining: Concepts and Techniques 17 Ordinal Variables An ordinal variable can be discrete or continuous Order is important, e.g., rank Can be treated like interval-scaled replace xif by their rank map the range of each variable onto [0, 1] by replacing i-th object in the f-th variable by zif rif {1,...,M f } rif 1 M f 1 compute the dissimilarity using methods for intervalscaled variables May 22, 2017 Data Mining: Concepts and Techniques 18 Ratio-Scaled Variables Ratio-scaled variable: a positive measurement on a nonlinear scale, approximately at exponential scale, such as AeBt or Ae-Bt Methods: treat them like interval-scaled variables—not a good choice! (why?—the scale can be distorted) apply logarithmic transformation yif = log(xif) treat them as continuous ordinal data treat their rank as interval-scaled May 22, 2017 Data Mining: Concepts and Techniques 19 Variables of Mixed Types A database may contain all the six types of variables symmetric binary, asymmetric binary, nominal, ordinal, interval and ratio One may use a weighted formula to combine their effects pf 1 ij( f ) d ij( f ) d (i, j) pf 1 ij( f ) f is binary or nominal: dij(f) = 0 if xif = xjf , or dij(f) = 1 otherwise f is interval-based: use the normalized distance f is ordinal or ratio-scaled compute ranks rif and r 1 z if and treat zif as interval-scaled M 1 if f May 22, 2017 Data Mining: Concepts and Techniques 20 Vector Objects Vector objects: keywords in documents, gene features in micro-arrays, etc. Broad applications: information retrieval, biologic taxonomy, etc. Cosine measure A variant: Tanimoto coefficient May 22, 2017 Data Mining: Concepts and Techniques 21 Chapter 7. Cluster Analysis 1. What is Cluster Analysis? 2. Types of Data in Cluster Analysis 3. A Categorization of Major Clustering Methods 4. Partitioning Methods 5. Hierarchical Methods 6. Density-Based Methods 7. Grid-Based Methods 8. Model-Based Methods 9. Clustering High-Dimensional Data 10. Constraint-Based Clustering 11. Outlier Analysis 12. Summary May 22, 2017 Data Mining: Concepts and Techniques 22 Major Clustering Approaches (I) Partitioning approach: Construct various partitions and then evaluate them by some criterion, e.g., minimizing the sum of square errors Typical methods: k-means, k-medoids Hierarchical approach: Create a hierarchical decomposition of the set of data (or objects) using some criterion Typical methods: Agnes, CAMELEON Density-based approach: Based on connectivity and density functions Typical methods: DBSACN, OPTICS, DenClue May 22, 2017 Data Mining: Concepts and Techniques 23 Major Clustering Approaches (II) Grid-based approach: based on a multiple-level granularity structure Typical methods: STING, WaveCluster, CLIQUE Model-based: A model is hypothesized for each of the clusters and tries to find the best fit of that model to each other Typical methods: EM, SOM, COBWEB Frequent pattern-based: Based on the analysis of frequent patterns Typical methods: pCluster User-guided or constraint-based: Clustering by considering user-specified or application-specific constraints Typical methods: COD (obstacles), constrained clustering May 22, 2017 Data Mining: Concepts and Techniques 24 Typical Alternatives to Calculate the Distance between Clusters Single link: smallest distance between an element in one cluster and an element in the other, i.e., dis(Ki, Kj) = min(tip, tjq) Complete link: largest distance between an element in one cluster and an element in the other, i.e., dis(Ki, Kj) = max(tip, tjq) Average: avg distance between an element in one cluster and an element in the other, i.e., dis(Ki, Kj) = avg(tip, tjq) Centroid: distance between the centroids of two clusters, i.e., dis(Ki, Kj) = dis(Ci, Cj) Medoid: distance between the medoids of two clusters, i.e., dis(Ki, Kj) = dis(Mi, Mj) Medoid: one chosen, centrally located object in the cluster May 22, 2017 Data Mining: Concepts and Techniques 25 Centroid, Radius and Diameter of a Cluster (for numerical data sets) Centroid: the “middle” of a cluster ip ) N Radius: square root of average distance from any point of the cluster to its centroid Cm iN 1(t N (t cm ) 2 Rm i 1 ip N Diameter: square root of average mean squared distance between all pairs of points in the cluster May 22, 2017 Data Mining: Concepts and Techniques 26 Chapter 7. Cluster Analysis 1. What is Cluster Analysis? 2. Types of Data in Cluster Analysis 3. A Categorization of Major Clustering Methods 4. Partitioning Methods 5. Hierarchical Methods 6. Density-Based Methods 7. Grid-Based Methods 8. Model-Based Methods 9. Clustering High-Dimensional Data 10. Constraint-Based Clustering 11. Outlier Analysis 12. Summary May 22, 2017 Data Mining: Concepts and Techniques 27 Partitioning Algorithms: Basic Concept Partitioning method: Construct a partition of a database D of n objects into a set of k clusters, s.t., min sum of squared distance k m1 tmiKm (Cm tmi ) 2 Given a k, find a partition of k clusters that optimizes the chosen partitioning criterion Global optimal: exhaustively enumerate all partitions Heuristic methods: k-means and k-medoids algorithms k-means (MacQueen’67): Each cluster is represented by the center of the cluster k-medoids or PAM (Partition around medoids) (Kaufman & Rousseeuw’87): Each cluster is represented by one of the objects in the cluster May 22, 2017 Data Mining: Concepts and Techniques 28 The K-Means Clustering Method Given k, the k-means algorithm is to partition objects into k nonempty subsets 0. Compute K initial centroids (randomly or using prior knowledge) 1. Assign each object to the cluster with the nearest centroids 2. Re-calculate the centroid of each cluster 3. Go back to Step 1, stop when no more new assignment May 22, 2017 Data Mining: Concepts and Techniques 29 The K-Means Clustering Method Example 10 10 9 9 8 8 7 7 6 6 5 5 10 9 8 7 6 5 4 4 3 2 1 0 0 1 2 3 4 5 6 7 8 K=2 Arbitrarily choose K object as initial cluster center 9 10 Assign each objects to most similar center 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 4 3 2 1 0 0 1 2 3 4 5 6 reassign 10 10 9 9 8 8 7 7 6 6 5 5 4 2 1 0 0 1 2 3 4 5 6 7 8 7 8 9 10 reassign 3 May 22, 2017 Update the cluster means 9 10 Update the cluster means Data Mining: Concepts and Techniques 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 30 Comments on the K-Means Method Strength: Relatively efficient: O(tkn), where n is # objects, k is # clusters, and t is # iterations. Normally, k, t << n. Comment: Often terminates at a local optimum. The global optimum may be found using techniques such as: genetic algorithms (how?) Weakness Applicable only when mean is defined, then what about categorical data? Need to specify k, the number of clusters, in advance Hard to handle noisy data and outliers May 22, 2017 Data Mining: Concepts and Techniques 31 Variations of the K-Means Method A few variants of the k-means which differ in Selection of the initial k means Dissimilarity calculations Strategies to calculate cluster means Handling categorical data: k-modes (Huang’98) Replacing means of clusters with modes Using new dissimilarity measures to deal with categorical objects Using a frequency-based method to update modes of clusters May 22, 2017 Data Mining: Concepts and Techniques 32 What Is the Problem of the K-Means Method? The k-means algorithm is sensitive to outliers ! Since an object with an extremely large value may substantially distort the distribution of the data. (Given an example?) K-Medoids: Instead of taking the mean value of the object in a cluster as a reference point, medoids can be used, which is the most centrally located object in a cluster. 10 10 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0 May 22, 2017 1 2 3 4 5 6 7 8 9 10 0 1 2 3 Data Mining: Concepts and Techniques 4 5 6 7 8 9 10 33 The K-Medoids Clustering Method Find representative objects, called medoids, in clusters PAM (Partitioning Around Medoids, 1987) starts from an initial set of medoids and iteratively replaces one of the medoids by one of the non-medoids if it improves the total distance of the resulting clustering PAM works effectively for small data sets, but does not scale well for large data sets May 22, 2017 Data Mining: Concepts and Techniques 34 A Typical K-Medoids Algorithm (PAM) Total Cost = 20 10 10 10 9 9 9 8 8 8 Arbitrary choose k object as initial medoids 7 6 5 4 3 2 7 6 5 4 3 2 1 1 0 0 0 1 2 3 4 5 6 7 8 9 0 10 1 2 3 4 5 6 7 8 9 10 Assign each remainin g object to nearest medoids 7 6 5 4 3 2 1 0 0 K=2 Until no change 10 3 4 5 6 7 8 9 10 10 Compute total cost of swapping 9 9 Swapping O and Oramdom 8 If quality is improved. 5 5 4 4 3 3 2 2 1 1 7 6 0 8 7 6 0 0 May 22, 2017 2 Randomly select a nonmedoid object,Oramdom Total Cost = 26 Do loop 1 1 2 3 4 5 6 7 8 9 10 Data Mining: Concepts and Techniques 0 1 2 3 4 5 6 7 8 9 10 35 PAM (Partitioning Around Medoids) (1987) PAM (Kaufman and Rousseeuw, 1987), built in Splus Use real object to represent the cluster Select k representative objects arbitrarily For each pair of non-selected object h and selected object i, calculate the total swapping cost Tcih For each pair of i and h, If TCih < 0, i is replaced by h Then assign each non-selected object to the most similar representative object repeat steps 2-3 until there is no change May 22, 2017 Data Mining: Concepts and Techniques 36 PAM Clustering: Total swapping cost TCih=jCjih 10 10 9 9 t 8 7 7 6 5 i 4 3 j 6 h 4 5 h i 3 2 2 1 1 0 0 0 1 2 3 4 5 6 7 8 9 10 Cjih = d(j, h) - d(j, i) May 22, 2017 j t 8 0 1 2 3 4 5 6 7 8 9 10 Cjih = 0 Data Mining: Concepts and Techniques 37 A Medoids Clustering Example May 22, 2017 Data Mining: Concepts and Techniques 38 Calculate Cost: May 22, 2017 Data Mining: Concepts and Techniques 39 May 22, 2017 Data Mining: Concepts and Techniques 40 Swap Medoids May 22, 2017 Data Mining: Concepts and Techniques 41 May 22, 2017 Data Mining: Concepts and Techniques 42 What Is the Problem with PAM? Pam is more robust than k-means in the presence of noise and outliers because a medoid is less influenced by outliers or other extreme values than a mean Pam works efficiently for small data sets but does not scale well for large data sets. O(k(n-k)2 ) for each iteration where n is # of data,k is # of clusters Sampling based method, CLARA(Clustering LARge Applications) May 22, 2017 Data Mining: Concepts and Techniques 43 CLARA (Clustering Large Applications) (1990) CLARA (Kaufmann and Rousseeuw in 1990) Built in statistical analysis packages, such as S+ It draws multiple samples of the data set, applies PAM on each sample, and gives the best clustering as the output Strength: deals with larger data sets than PAM Weakness: Efficiency depends on the sample size A good clustering based on samples will not necessarily represent a good clustering of the whole data set if the sample is biased May 22, 2017 Data Mining: Concepts and Techniques 44 Chapter 7. Cluster Analysis 1. What is Cluster Analysis? 2. Types of Data in Cluster Analysis 3. A Categorization of Major Clustering Methods 4. Partitioning Methods 5. Hierarchical Methods 6. Density-Based Methods 7. Grid-Based Methods 8. Model-Based Methods 9. Clustering High-Dimensional Data 10. Constraint-Based Clustering 11. Outlier Analysis 12. Summary May 22, 2017 Data Mining: Concepts and Techniques 45 Hierarchical Clustering Use distance matrix as clustering criteria. This method does not require the number of clusters k as an input, but needs a termination condition Step 0 a Step 1 Step 2 Step 3 Step 4 agglomerative (AGNES) ab b abcde c cde d de e Step 4 May 22, 2017 Step 3 Step 2 Step 1 Step 0 Data Mining: Concepts and Techniques divisive (DIANA) 46 AGNES (Agglomerative Nesting) Introduced in Kaufmann and Rousseeuw (1990) Implemented in statistical analysis packages, e.g., Splus Use the Single-Link method and the dissimilarity matrix. Merge nodes that have the least dissimilarity Eventually all nodes belong to the same cluster 10 10 10 9 9 9 8 8 8 7 7 7 6 6 6 5 5 5 4 4 4 3 3 3 2 2 2 1 1 1 0 0 0 1 2 May 22, 2017 3 4 5 6 7 8 9 10 0 0 1 2 3 4 5 6 7 8 9 10 Data Mining: Concepts and Techniques 0 1 2 3 4 5 6 7 8 9 10 47 Dendrogram: Shows How the Clusters are Merged Decompose data objects into a several levels of nested partitioning (tree of clusters), called a dendrogram. May 22, 2017 Data Mining: Concepts and Techniques 48 DIANA (Divisive Analysis) Introduced in Kaufmann and Rousseeuw (1990) Implemented in statistical analysis packages, e.g., Splus Inverse order of AGNES Eventually each node forms a cluster on its own 10 10 10 9 9 9 8 8 8 7 7 7 6 6 6 5 5 5 4 4 4 3 3 3 2 2 2 1 1 1 0 0 0 0 1 2 May 22, 2017 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Data Mining: Concepts and Techniques 0 1 2 3 4 5 6 7 8 9 10 49 Recent Hierarchical Clustering Methods Major weakness of agglomerative clustering methods do not scale well: time complexity of at least O(n2), where n is the number of total objects can never undo what was done previously Integration of hierarchical with distance-based clustering CHAMELEON (1999): hierarchical clustering using dynamic modeling May 22, 2017 Data Mining: Concepts and Techniques 50 Overall Framework of CHAMELEON Construct Partition the Graph Sparse Graph Data Set Merge Partition Final Clusters Implemented in http://glaros.dtc.umn.edu/gkhome/views/cluto May 22, 2017 Data Mining: Concepts and Techniques 51 CHAMELEON (Clustering Complex Objects) May 22, 2017 Data Mining: Concepts and Techniques 52 Chapter 7. Cluster Analysis 1. What is Cluster Analysis? 2. Types of Data in Cluster Analysis 3. A Categorization of Major Clustering Methods 4. Partitioning Methods 5. Hierarchical Methods 6. Density-Based Methods 7. Grid-Based Methods 8. Model-Based Methods 9. Clustering High-Dimensional Data 10. Constraint-Based Clustering 11. Outlier Analysis 12. Summary May 22, 2017 Data Mining: Concepts and Techniques 53 Density-Based Clustering Methods Clustering based on density (local cluster criterion), such as density-connected points Major features: Discover clusters of arbitrary shape Handle noise One scan Need density parameters as termination condition Several interesting studies: DBSCAN: Ester, et al. (KDD’96) OPTICS: Ankerst, et al (SIGMOD’99). DENCLUE: Hinneburg & D. Keim (KDD’98) May 22, 2017 Data Mining: Concepts and Techniques 54 Density-Based Clustering: Basic Concepts Two parameters: Eps: Maximum radius of the neighbourhood MinPts: Minimum number of points in an Epsneighbourhood of that point NEps(p): {q belongs to D | dist(p,q) <= Eps} Directly density-reachable: A point p is directly densityreachable from a point q w.r.t. Eps, MinPts if p belongs to NEps(q) core point condition: |NEps (q)| >= MinPts May 22, 2017 Data Mining: Concepts and Techniques p q MinPts = 5 Eps = 1 cm 55 Density-Reachable and Density-Connected Density-reachable: A point p is density-reachable from a point q w.r.t. Eps, MinPts if there is a chain of points p1, …, pn, p1 = q, pn = p such that pi+1 is directly density-reachable from pi p p1 q Density-connected A point p is density-connected to a point q w.r.t. Eps, MinPts if there is a point o such that both, p and q are density-reachable from o w.r.t. Eps and MinPts May 22, 2017 p Data Mining: Concepts and Techniques q o 56 DBSCAN: Density Based Spatial Clustering of Applications with Noise Relies on a density-based notion of cluster: A cluster is defined as a maximal set of density-connected points Discovers clusters of arbitrary shape in spatial databases with noise Outlier Border Eps = 1cm Core May 22, 2017 MinPts = 5 Data Mining: Concepts and Techniques 57 DBSCAN: The Algorithm Arbitrary select a point p Retrieve all points density-reachable from p w.r.t. Eps and MinPts. If p is a core point, a cluster is formed. If p is a border point, no points are density-reachable from p and DBSCAN visits the next point of the database. Continue the process until all of the points have been processed. May 22, 2017 Data Mining: Concepts and Techniques 58 DBSCAN: Sensitive to Parameters May 22, 2017 Data Mining: Concepts and Techniques 59 Density-Based Clustering: OPTICS & Its Applications May 22, 2017 Data Mining: Concepts and Techniques 60 DENCLUE: Using Statistical / Probability Density Functions DENsity-based CLUstEring by Hinneburg & Keim (KDD’98) 2 Using statistical density functions: f Gaussian ( x, y) e f D Gaussian f Major features ( x) d ( x,y) 2 2 N i 1 e d ( x , xi ) 2 2 2 ( x, xi ) i 1 ( xi x) e D Gaussian Solid mathematical foundation Good for data sets with large amounts of noise N d ( x , xi ) 2 2 2 Allows a compact mathematical description of arbitrarily shaped clusters in high-dimensional data sets Significant faster than existing algorithm (e.g., DBSCAN) But needs a large number of parameters May 22, 2017 Data Mining: Concepts and Techniques 61 Denclue: Technical Essence Influence function: describes the impact of a data point within its neighborhood Overall density of the data space can be calculated as the sum of the influence function of all data points Clusters can be determined mathematically by identifying density attractors Density attractors are local maximal of the overall density function May 22, 2017 Data Mining: Concepts and Techniques 62 Density Attractor May 22, 2017 Data Mining: Concepts and Techniques 63 Hill Climbing Clustering Hinneburg and Keim, 1994 May 22, 2017 Data Mining: Concepts and Techniques 64 Handle Noise and Outliers May 22, 2017 Data Mining: Concepts and Techniques 65 May 22, 2017 Data Mining: Concepts and Techniques 66 Center-Defined and Arbitrary May 22, 2017 Data Mining: Concepts and Techniques 67 Chapter 7. Cluster Analysis 1. What is Cluster Analysis? 2. Types of Data in Cluster Analysis 3. A Categorization of Major Clustering Methods 4. Partitioning Methods 5. Hierarchical Methods 6. Density-Based Methods 7. Grid-Based Methods 8. Model-Based Methods 9. Clustering High-Dimensional Data 10. Constraint-Based Clustering 11. Outlier Analysis 12. Summary May 22, 2017 Data Mining: Concepts and Techniques 68 May 22, 2017 Data Mining: Concepts and Techniques 69 May 22, 2017 Data Mining: Concepts and Techniques 70 Grid-Based Clustering Method Using multi-resolution grid data structure Several interesting methods STING (a STatistical INformation Grid approach) by Wang, Yang and Muntz (1997) WaveCluster by Sheikholeslami, Chatterjee, and Zhang (VLDB’98) May 22, 2017 A multi-resolution clustering approach using wavelet method Data Mining: Concepts and Techniques 71 STING: A Statistical Information Grid Approach Wang, Yang and Muntz (VLDB’97) The spatial area area is divided into rectangular cells There are several levels of cells corresponding to different levels of resolution May 22, 2017 Data Mining: Concepts and Techniques 72 The STING Clustering Method Each cell at a high level is partitioned into a number of smaller cells in the next lower level Statistical info of each cell is calculated and stored beforehand and is used to answer queries Parameters of higher level cells can be easily calculated from parameters of lower level cell count, mean, s, min, max type of distribution—normal, uniform, etc. Use a top-down approach to answer spatial data queries Start from a pre-selected layer—typically with a small number of cells For each cell in the current level compute the confidence interval May 22, 2017 Data Mining: Concepts and Techniques 73 Top Down Search May 22, 2017 Data Mining: Concepts and Techniques 74 May 22, 2017 Data Mining: Concepts and Techniques 75 Comments on STING Remove the irrelevant cells from further consideration When finish examining the current layer, proceed to the next lower level Repeat this process until the bottom layer is reached Advantages: Query-independent, easy to parallelize, incremental update O(K), where K is the number of grid cells at the lowest level Disadvantages: All the cluster boundaries are either horizontal or vertical, and no diagonal boundary is detected May 22, 2017 Data Mining: Concepts and Techniques 76 WaveCluster A multi-resolution clustering approach which applies wavelet transform to the feature space A wavelet transform is a signal processing technique that composes a signal into different frequency sub-band. Both grid-based and density-based Input parameters: # of grid cells for each dimension the wavelet, and the # of applications of wavelet transform. WaveCluster How to apply wavelet transform to find clusters Summaries the data by imposing a multidimensional grid structure onto data space These multidimensional spatial data objects are represented in an n-dimensional feature space Apply wavelet transform on feature space to find the dense regions in the feature space Apply wavelet transform multiple times which result in clusters at different scales from fine to coarse Wavelet Transform Wavelet transform: A signal processing technique that decomposes a signal into different frequency interval / sub-band Data are transformed to preserve relative distance between objects at different levels of resolution Allows natural clusters to become more distinguishable May 22, 2017 Data Mining: Concepts and Techniques 79 Quantization May 22, 2017 Data Mining: Concepts and Techniques 81 Transformation and Clustering WaveCluster Why is wavelet transformation useful for clustering Unsupervised clustering It uses hat-shape filters to emphasize region where points cluster, but simultaneously to suppress weaker information in their boundary WaveCluster Effective removal of outliers WaveCluster Remove Noise and Identify Complicated Clusters May 22, 2017 Data Mining: Concepts and Techniques 86 The WaveCluster Algorithm Input parameters # of grid cells for each dimension the wavelet, and the # of applications of wavelet transform Why is wavelet transformation useful for clustering? Use hat-shape filters to emphasize region where points cluster, but simultaneously suppress weaker information in their boundary Effective removal of outliers, multi-resolution, cost effective Major features: Complexity O(N) Detect arbitrary shaped clusters at different scales Not sensitive to noise, not sensitive to input order Only applicable to low dimensional data Both grid-based and density-based May 22, 2017 Data Mining: Concepts and Techniques 87 May 22, 2017 Data Mining: Concepts and Techniques 88 Chapter 7. Cluster Analysis 1. What is Cluster Analysis? 2. Types of Data in Cluster Analysis 3. A Categorization of Major Clustering Methods 4. Partitioning Methods 5. Hierarchical Methods 6. Density-Based Methods 7. Grid-Based Methods 8. Model-Based Methods 9. Clustering High-Dimensional Data 10. Constraint-Based Clustering 11. Outlier Analysis 12. Summary May 22, 2017 Data Mining: Concepts and Techniques 89 Model-Based Clustering What is model-based clustering? Attempt to optimize the fit between the given data and some mathematical model Based on the assumption: Data are generated by a mixture of underlying probability distribution Typical methods Statistical approach EM (Expectation maximization) Neural network approach May 22, 2017 SOM (Self-Organizing Feature Map) Data Mining: Concepts and Techniques 90 EM — Expectation Maximization EM — A popular iterative refinement algorithm EM clustering is a soft clustering in contrast to k-means hard clustering New means are computed based on weighted measures General idea Assign each object to a cluster according to a probability distribution (weight) Starts with an initial estimate of the parameter vector of each cluster Iteratively rescores the patterns (data points) against the mixture density produced by the parameter vector The rescored patterns are used to update the parameter updates Patterns belonging to the same cluster, if they are placed by their scores in a particular component Algorithm converges fast but may not be in global optima May 22, 2017 Data Mining: Concepts and Techniques 91 The EM (Expectation Maximization) Algorithm Initially, randomly assign k cluster centers / parameters Iteratively refine the clusters based on two steps Expectation step: assign each data point Xi to cluster Ci with the following probability Maximization step: Estimation of model parameters May 22, 2017 Data Mining: Concepts and Techniques 92 Multivariate Gaussian Distribution for P(X | C) How to re-estimate parameters? May 22, 2017 Data Mining: Concepts and Techniques 93 Naïve Bayes Clustering Data: X1, X2, …, Xn Attributes: A1, A2, …, Ad Clusters: C1, C2, …, Ck Initialize a model P(Ai = Vm | Cj), 1 <= i <= d, 1 <= j <= k, 1<= m <= M P(Cj): proportion of data in Cj, 1 <= j <= k May 22, 2017 Data Mining: Concepts and Techniques 94 Naïve Bayes Clustering E-Step: soft assignment Calculate P(Cj | Xi) = P(Xi | Cj) * P(Cj) / P(Xi) M-Step: re-estimate parameters P(Cj) = ∑ P(Cj | Xi) / N P(Ak = Vm| Cj ) = (∑ P(Cj | Xi) * δ(Ak of Xi is Vm)) / ∑ P(Cj | Xi) Repeat E- and M- Steps until it converges May 22, 2017 Data Mining: Concepts and Techniques 95 Neural Network Approach Neural network approaches Represent each cluster as an exemplar, acting as a “prototype” of the cluster New objects are distributed to the cluster whose exemplar is the most similar according to some distance measure Typical methods SOM (Soft-Organizing feature Map) Competitive learning May 22, 2017 Involves a grid architecture of several units (neurons) Neurons compete in a “winner-takes-all” fashion for the object currently being presented Data Mining: Concepts and Techniques 96 Self-Organizing Feature Map (SOM) SOMs, also called topological ordered maps, or Kohonen Self-Organizing Feature Map (KSOMs) It maps all the points in a high-dimensional source space into a 2 to 3-d target space, s.t., the distance and proximity relationship (i.e., topology) are preserved as much as possible Similar to k-means: cluster centers tend to lie in a low-dimensional manifold in the feature space Clustering is performed by having several units competing for the current object The unit whose weight vector is closest to the current object wins The winner and its neighbors learn by having their weights adjusted SOMs are believed to resemble processing that can occur in the brain Useful for visualizing high-dimensional data in 2- or 3-D space May 22, 2017 Data Mining: Concepts and Techniques 97 May 22, 2017 Data Mining: Concepts and Techniques 98 May 22, 2017 Data Mining: Concepts and Techniques 99 May 22, 2017 Data Mining: Concepts and Techniques 100 May 22, 2017 Data Mining: Concepts and Techniques 101 Web Document Clustering Using SOM The result of SOM clustering of 12088 Web articles The picture on the right: drilling down on the keyword “mining” Based on websom.hut.fi Web page May 22, 2017 Data Mining: Concepts and Techniques 102 Chapter 6. Cluster Analysis 1. What is Cluster Analysis? 2. Types of Data in Cluster Analysis 3. A Categorization of Major Clustering Methods 4. Partitioning Methods 5. Hierarchical Methods 6. Density-Based Methods 7. Grid-Based Methods 8. Model-Based Methods 9. Clustering High-Dimensional Data 10. Constraint-Based Clustering 11. Outlier Analysis 12. Summary May 22, 2017 Data Mining: Concepts and Techniques 103 Clustering High-Dimensional Data Clustering high-dimensional data Many applications: text documents, DNA micro-array data Major challenges: Many irrelevant dimensions may mask clusters Distance measure becomes meaningless—due to equi-distance Clusters may exist only in some subspaces Methods Feature transformation: only effective if most dimensions are relevant Feature selection: wrapper or filter approaches PCA & SVD useful only when features are highly correlated/redundant useful to find a subspace where the data have nice clusters Subspace-clustering: find clusters in all the possible subspaces May 22, 2017 CLIQUE and frequent pattern-based clustering Data Mining: Concepts and Techniques 104 The Curse of Dimensionality (graphs adapted from Parsons et al. KDD Explorations 2004) Data in only one dimension is relatively packed Adding a dimension “stretch” the points across that dimension, making them further apart Adding more dimensions will make the points further apart—high dimensional data is extremely sparse Distance measure becomes meaningless—due to equi-distance May 22, 2017 Data Mining: Concepts and Techniques 105 Why Subspace Clustering? (adapted from Parsons et al. SIGKDD Explorations 2004) May 22, 2017 Clusters may exist only in some subspaces Subspace-clustering: find clusters in all the subspaces Data Mining: Concepts and Techniques 106 CLIQUE (Clustering In QUEst) Agrawal, Gehrke, Gunopulos, Raghavan (SIGMOD’98) Automatically identifying subspaces of a high dimensional data space that allow better clustering than original space CLIQUE can be considered as both density-based and grid-based It partitions each dimension into the same number of equal length interval It partitions an m-dimensional data space into non-overlapping rectangular units A unit is dense if the fraction of total data points contained in the unit exceeds the input model parameter A cluster is a maximal set of connected dense units within a subspace May 22, 2017 Data Mining: Concepts and Techniques 107 CLIQUE: The Major Steps Partition the data space and find the number of points that lie inside each cell of the partition. Identify clusters Determine dense units in all subspaces of interests Determine connected dense units in all subspaces of interests. Generate minimal description for the clusters Determine maximal regions that cover a cluster of connected dense units for each cluster Determination of minimal cover for each cluster May 22, 2017 Data Mining: Concepts and Techniques 108 40 50 20 30 40 50 age 60 Vacation =3 30 Vacation (week) 0 1 2 3 4 5 6 7 Salary (10,000) 0 1 2 3 4 5 6 7 20 age 60 30 50 age May 22, 2017 Data Mining: Concepts and Techniques 109 Strength and Weakness of CLIQUE Strength automatically finds subspaces of the highest dimensionality such that high density clusters exist in those subspaces insensitive to the order of records in input and does not presume some canonical data distribution scales linearly with the size of input and has good scalability as the number of dimensions in the data increases Weakness The accuracy of the clustering result may be degraded at the expense of simplicity of the method May 22, 2017 Data Mining: Concepts and Techniques 110 Frequent Pattern-Based Approach Clustering high-dimensional space (e.g., clustering text documents, microarray data) Projected subspace-clustering: which dimensions to be projected on? CLIQUE Using frequent patterns as “features” “Frequent” are inherent features Mining freq. patterns may not be so expensive Typical methods Frequent-term-based document clustering Clustering by pattern similarity in micro-array data (pClustering) May 22, 2017 Data Mining: Concepts and Techniques 111 Clustering by Pattern Similarity (p-Clustering) Right: The micro-array “raw” data shows 3 genes and their values in a multi-dimensional space Difficult to find their patterns Bottom: Some subsets of dimensions form nice shift and scaling patterns May 22, 2017 Data Mining: Concepts and Techniques 112 Why p-Clustering? Microarray data analysis may need to Clustering on thousands of dimensions (attributes) Discovery of both shift and scaling patterns Clustering with Euclidean distance measure? — cannot find shift patterns Clustering on derived attribute Aij = ai – aj? — introduces N(N-1) dimensions Bi-cluster using transformed mean-squared residue score matrix (I, J) d 1 d | J | j J ij d 1 d | I | i I ij d 1 d | I || J | i I , j J ij Where A submatrix is a δ-cluster if H(I, J) ≤ δ for some δ > 0 iJ Ij IJ Problems with bi-cluster No downward closure property, Due to averaging, it may contain outliers but still within δ-threshold May 22, 2017 Data Mining: Concepts and Techniques 113 H(I, J) Matrix of Bi-Clustering J I i j dij dIj May 22, 2017 Data Mining: Concepts and Techniques diJ dIJ 114 H(I, J) Matrix of Bi-Clustering J I i j dij-dIj – diJ + dIJ dIj May 22, 2017 Data Mining: Concepts and Techniques diJ dIJ 115 p-Clustering: Clustering by Pattern Similarity Given object x, y in O and features a, b in T, pCluster is a 2 by 2 matrix d xa d xb pScore( ) | (d xa d xb ) (d ya d yb ) | d ya d yb A pair (O, T) is in δ-pCluster if for any 2 by 2 matrix X in (O, T), pScore(X) ≤ δ for some δ > 0 Properties of δ-pCluster Downward closure Clusters are more homogeneous than bi-cluster (thus the name: pair-wise Cluster) Pattern-growth algorithm has been developed for efficient mining d /d ya For scaling patterns, one can observe, taking logarithmic on xa d xb / d yb will lead to the pScore form May 22, 2017 Data Mining: Concepts and Techniques 116 Chapter 6. Cluster Analysis 1. What is Cluster Analysis? 2. Types of Data in Cluster Analysis 3. A Categorization of Major Clustering Methods 4. Partitioning Methods 5. Hierarchical Methods 6. Density-Based Methods 7. Grid-Based Methods 8. Model-Based Methods 9. Clustering High-Dimensional Data 10. Constraint-Based Clustering 11. Outlier Analysis 12. Summary May 22, 2017 Data Mining: Concepts and Techniques 117 Why Constraint-Based Cluster Analysis? Need user feedback: Users know their applications the best Less parameters but more user-desired constraints, e.g., an ATM allocation problem: obstacle & desired clusters May 22, 2017 Data Mining: Concepts and Techniques 118 A Classification of Constraints in Cluster Analysis Clustering in applications: desirable to have user-guided (i.e., constrained) cluster analysis Different constraints in cluster analysis: Constraints on individual objects (do selection first) Constraints on distance or similarity functions # of clusters, MinPts, etc. User-specified constraints Weighted functions, obstacles (e.g., rivers, lakes) Constraints on the selection of clustering parameters Cluster on houses worth over $300K Contain at least 500 valued customers and 5000 ordinary ones Semi-supervised: giving small training sets as “constraints” or hints May 22, 2017 Data Mining: Concepts and Techniques 119 An Example: Clustering With Obstacle Objects Not Taking obstacles into account May 22, 2017 Taking obstacles into account Data Mining: Concepts and Techniques 120 Chapter 7. Cluster Analysis 1. What is Cluster Analysis? 2. Types of Data in Cluster Analysis 3. A Categorization of Major Clustering Methods 4. Partitioning Methods 5. Hierarchical Methods 6. Density-Based Methods 7. Grid-Based Methods 8. Model-Based Methods 9. Clustering High-Dimensional Data 10. Constraint-Based Clustering 11. Outlier Analysis 12. Summary May 22, 2017 Data Mining: Concepts and Techniques 121 What Is Outlier Discovery? What are outliers? The set of objects are considerably dissimilar from the remainder of the data Example: Sports: Michael Jordon, Wayne Gretzky, ... Problem: Define and find outliers in large data sets Applications: Credit card fraud detection Telecom fraud detection Customer segmentation Medical analysis Bioinformatics May 22, 2017 Data Mining: Concepts and Techniques 122 Outlier Discovery: Statistical Approaches Assume a model underlying distribution that generates data set (e.g. normal distribution) Use discordancy tests depending on data distribution distribution parameter (e.g., mean, variance) number of expected outliers Drawbacks most tests are for single attribute In many cases, data distribution may not be known May 22, 2017 Data Mining: Concepts and Techniques 123 Outlier Discovery: Distance-Based Approach Introduced to counter the main limitations imposed by statistical methods We need multi-dimensional analysis without knowing data distribution Distance-based outlier: A DB(p, d)-outlier is an object O in a dataset T such that at least a fraction p of the objects in T lies at a distance greater than d from O May 22, 2017 Data Mining: Concepts and Techniques 124 Density-Based Local Outlier Detection Distance-based outlier detection is based on global distance distribution It encounters difficulties to identify outliers if data is not uniformly distributed Ex. C1 contains 400 loosely distributed points, C2 has 100 tightly condensed points, 2 outlier points o1, o2 Distance-based method cannot identify o2 as an outlier Need the concept of local outlier May 22, 2017 Data Mining: Concepts and Techniques 125 Outlier Discovery: Deviation-Based Approach Identifies outliers by examining the main characteristics of objects in a group Objects that “deviate” from this description are considered outliers Sequential exception technique simulates the way in which humans can distinguish unusual objects from among a series of supposedly like objects Data cube technique uses data cubes to identify regions of anomalies in large multidimensional data May 22, 2017 Data Mining: Concepts and Techniques 126 Chapter 7. Cluster Analysis 1. What is Cluster Analysis? 2. Types of Data in Cluster Analysis 3. A Categorization of Major Clustering Methods 4. Partitioning Methods 5. Hierarchical Methods 6. Density-Based Methods 7. Grid-Based Methods 8. Model-Based Methods 9. Clustering High-Dimensional Data 10. Constraint-Based Clustering 11. Outlier Analysis 12. Summary May 22, 2017 Data Mining: Concepts and Techniques 127 Summary Cluster analysis groups objects based on their similarity and has wide applications Measure of similarity can be computed for various types of data Clustering algorithms can be categorized into partitioning methods, hierarchical methods, density-based methods, grid-based methods, model-based methods, frequent pattern based method Outlier detection and analysis are very useful for fraud detection, etc. and can be performed by statistical, distance-based or deviation-based approaches There are still lots of research issues on cluster analysis May 22, 2017 Data Mining: Concepts and Techniques 128 Problems and Challenges Considerable progress has been made in scalable clustering methods Partitioning: k-means, k-medoids, CLARANS Hierarchical: BIRCH, ROCK, CHAMELEON Density-based: DBSCAN, OPTICS, DenClue Grid-based: STING, WaveCluster, CLIQUE Model-based: EM, Cobweb, SOM Frequent pattern-based: pCluster Constraint-based: COD, constrained-clustering Current clustering techniques do not address all the requirements adequately, still an active area of research May 22, 2017 Data Mining: Concepts and Techniques 129 References (1) R. 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